1. Avoiding Monochromatic Giants in Edge-Colorings of Random Graphs Henning Thomas (joint with Reto Spöhel, Angelika Steger) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

2. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Phase Transition of the Random Graph  [Erd ő s, Rényi (1960)] The random graph G n,cn whp. consists of Giant c < 0.5c > 0.5 - components of size o(n)if c < 0.5 - a single giant component of size ­(n) and other components of size o(n) if c > 0.5

3. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Achlioptas Process  [Bohman, Frieze (2001)],..., [Spencer, Wormald (2007)] In the Achlioptas process the emergence of the giant component can be slowed down or accelerated by a constant factor.  No exact thresholds are known; current best bounds are: [Spencer, Wormald (2007)]: Whp. a giant component can be  avoided for at least 0.829n edge pairs,  created within 0.334n edge pairs.

4. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Corresponding Offline Problem  Given n vertices and cn random edge pairs, is it possible to select one edge from every pair such that in the resulting graph every component has size o(n) ?  [Bohman, Kim (2006)] This property has a threshold at c 1 n for some analytically computable constant c 1 ¼ 0.9768. Unrestricted variant ([Bohman, Frieze, Wormald (2004)]): Given n vertices and 2cn random edges, is it possible to select cn edges such that in the resulting graph every component has size o ( n )? This property has a (slightly higher!) threshold at 2c 2 n for some analytically computable constant c 2 ¼ 0.9792.

5. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Coloring Variant of the Problem  Given n vertices and cn random edge pairs, is it possible to find a valid 2 -edge-coloring such that every monochromatic component has size o(n) ?  Valid: Both colors are used exactly once in every edge pair.

6. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Coloring Variant of the Problem  Let r ¸ 2 be fixed. Given n vertices and cn random r -sets of edges, is it possible to find a valid r - edge-coloring such that every monochromatic component has size o(n) ?  Valid: Each of the r colors is used exactly once in every r -set.

7. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Coloring Variant of the Problem  Let r ¸ 2 be fixed. Given n vertices and cn random r -sets of edges, is it possible to find a valid r - edge-coloring such that every monochromatic component has size o(n) ?  Valid: Each of the r colors is used exactly once in every r -set. r = 4

8. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Coloring Variant of the Problem  Let r ¸ 2 be fixed. Given n vertices and cn random r -sets of edges, is it possible to find a valid r - edge-coloring such that every monochromatic component has size o(n) ?  Valid: Each of the r colors is used exactly once in every r -set.  Theorem [Spöhel, Steger, T.] For every r ¸ 2 the property has a threshold at c r * n for some analytically computable constant c r *.  The threshold coincides with the threshold for r - orientability of the random graph G n,rcn. Unrestricted variant (ind. [Bohman, Frieze, Krivelevich, Loh, Sudakov]): Given n vertices and rcn random edges, is it possible to find an r - edge-coloring such that every monochromatic component has size o(n) ? This property has the same threshold as the restricted variant!

9. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 r -Orientability  G is r -orientable if there exists an orientation such that the in- degree of every vertex is at most r.  In fact, G is r -orientable iff m(G) · r, where m(G) := max H µ G e(H)=v(H) is the max. edge density of G.  The threshold for r -orientability of the random graph G n,m was determined by [Fernholz, Ramachandran (SODA 07)] and independently by [Cain, Sanders, Wormald (SODA 07)].  Setting m = rcn the threshold is at rc r * n. r23456789 cr*cr* 0.8820.9590.9800.9890.9940.9960.9980.999

10. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Upper Bound Proof  Let c > c r *. Need to show: Whp. every valid r -edge-coloring of cn random r -sets of edges contains a monochromatic giant.  We sample edges without replacement.  ) G := “  r -sets” is distributed like G n,rcn  Density Lemma ([Bohman, Frieze, Wormald (2004)]) Whp. All subgraphs in G of edge density ¸ 1+² have linear size.  Whp. we have m(G) ¸ (1+²)r  ) 9 subgraph with edge density ¸ (1+²)r  ) Every valid r -edge-coloring of G contains a monochromatic (connected!) subgraph with edge density ¸ 1+².

11. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Lower Bound Proof - Idea  Let c < c r *. Need to show: Whp. there exists a valid r -edge- coloring of cn random r -sets of edges in which every monochromatic component has size o(n).  “Inverse Two Round Exposure”:  We generate cn random r -sets by first generating (c+²)n random r -sets (with c+² < c r * ) and then deleting ²n random r -sets.  Let G + be the union of the (c+²)n r -sets (distributed like G n,r(c+²)n ).

12. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Lower Bound Proof - Outline  How to use this idea (inspired by [Bohman, Kim (2007)]):  First Round: Find a valid r -edge-coloring of G + in which every monochromatic component is low-connected  Second Round: Show that the edge deletion breaks the low- connected components into small ones.

13. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Lower Bound – First Round  Fact: The chromatic index of a bipartite graph G equals ¢(G)  This yields a valid r-edge-coloring of E(G + ) such that in every color class every vertex has in-degree at most 1.  ) Every monochromatic component is unicyclic or a tree. 2 1 5 34 2 1 5 34 1 2 3 4 5 B G+G+ V(G+)V(G+) r -sets Every edge - belongs to one r -set - points to one vertex 1 2 3 4 5 ¢(B) = r¢(B) = r 2 1 5 34 r = 2

14. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Lower Bound – Second Round (Sketch)  Consider a fixed color class with components C 1 +, …, C s +  Remove one edge from every cycle  Lemma: Deleting ²n random r -sets breaks the resulting trees into components of size o(n).  Then: Every component C i + breaks into components of size at most 2o(n) = o(n).

15. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Summary  Avoiding monochromatic giants in edge-colorings of random graphs has the same threshold as orientability of random graphs.  No difference between balanced and unbalanced setting (in contrast to edge-selection problems)  Related Work  Online setup  Creating the giant  Open Questions  Vertex-Coloring [Bohman, Frieze, Krivelevich, Loh, Sudakov] Po-Shen‘s Talk

16. Henning ThomasAvoiding Monochromatic Giants in Edge-Colorings of Random GraphsETH Zurich 2009 Lower Bound – Second Round (Sketch)  A vertex ‘survives’ if its first log(n)/K ancestors are not deleted  Pr[u survives] · (1-²/(c+²)) log(n)/K · n - ²/(c + ²)K  E[#surviving vertices]=O(n 1-²/(c+²)K )  Markov: Whp. #surviving vertices=o(n) (*)  Lemma ([Bohman, Kim (2007)]): Whp. all trees in G n,rcn with depth at most log(n)/K have o(n) vertices.(**)  Consider a tree after the edge deletion  Conditioning on (*) and (**) such a tree has o(n) vertices log(n)/K Tree of depth at most log(n)/K Surviving vertices